MULTIPLICATION OPERATORS IN MEASURABLE SECTIONS SPACES

Abstract: In this article we consider linear operators in measurable section spaces. Let X be a liftable measurable bundle of Banach spaces and E be an order continuous Köthe function space over a finite measure space (A,Σ, μ). We prove that a linear continuous operator T in a measurable sections space E(X ) is a multiplication operator (by a function in L∞(μ)) if and only if the equality T (g〈f, φ〉φ) = g〈T (f), φ〉φ) holds for every g ∈ L∞(μ), f ∈ E(X ), φ ∈ L∞(X ) and φ ⋆ ∈ L∞(X ).